Is There A Better Way To Simulate Pid Control In Python With Scipy's Solve_ivp()?

I am working on a homework problem. I'm trying to simulate a PID control in Python with Scipy's integrate.solve_ivp() function. My method is to run the PID code within the right-ha

Solution 1:

Let's look at a more simple system, the ubiquitous spring with dampening

    y'' + c*y' + k*y = u(t)

where u(t) could represent the force exerted by an electro-magnet (which immediately suggests ways to make the system more complicated by introducing a more realistic relation of voltage and magnetic field).

Now in a PID controller we have the error to a reference output e = yr - y and

    u(t) = kD*e'(t) + kP*e(t) + kI*integral(e(t))

To treat this with an ODE solver we immediately see that the state needs to be extended with a new component E(t) where E'(t)=e(t). The next difficulty is to implement the derivative of an expression that is not necessarily differentiable. This is achievable by avoiding to differentiate this expression at all by using a non-standard first-order implementation (where the standard would be to use [y,y',E] as state).

Essentially, collect all derived expressions in the formula in their integrated form as

    v(t)=y'(t)+c*y-kD*e(t). 

Then going back to the derivative one gets the first order system

    v'(t) = y''(t) + c*y'(t) - kD*e'(t) 
          = kP*e(t) + kI*E(t) - k*y(t)

    y'(t) = v(t) - c*y(t) + kD*e(t)
    E'(t) = e(t)

This now allows to implement the controlled system as ODE system without tricks involving a global memory or similar

defodePID(t,u,params):
    c,k,kD,kP,kI = params
    y,v,E = u
    e = yr(t)-y
    return [ v-c*y+kD*e, kP*e+kI*E-k*y, e ]

You should be able to use similar transformations of the first order system in your more complex model.